Exercise 15 Page 716

We are given a table that shows the lengths of four different boards.

Length (ft)
$12$	$15$	$15$	$20$

Also, we know that standard deviation is about $2.9$ feet. We are asked to determine which of the given statements is true.

Option	Statement	True or False?
F	The standard deviation is less than the mean absolute deviation.	?
G	The standard deviation is greater than the mean absolute deviation.	?
H	The mean absolute deviation is greater than the range.	?
I	The range is less than the standard deviation.	?

Let's first analyze statements from options F and G.

F and G

In options F and G, we are given the following statements.

Option	Statement	True or False?
F	The standard deviation is less than the mean absolute deviation.	?
G	The standard deviation is greater than the mean absolute deviation.	?

To determine whether these statements are true or false, we will need to calculate the mean absolute deviation. First, let's calculate the mean of the lengths of the boards from the table.

Lengths	$12$	$15$	$15$	$20$
Sum	$12+15+15+20=62$
Mean	$\frac{62}{4}=15.5$

We found the mean! Next, let's find the distance of each value from the mean. The distance is equal to the absolute value of the difference between each value in the data set and the mean.

Length	$|\text{Mean}-\text{Length}|$	Distance from the Mean
$12$	$|15.5-12|$	$3.5$
$15$	$|15.5-15|$	$0.5$
$15$	$|15.5-15|$	$0.5$
$20$	$|15.5-20|$	$4.5$

The mean absolute deviation is the mean of all the distances. To find it, we will first sum all the distances. Now let's divide the sum by the number of the distances, which is $4.$ We found that the mean absolute deviation is $2$ feet. Since the standard deviation is about $2.9$ feet, the standard deviation is greater than the mean absolute deviation. Therefore, the statement from option G is true and the statement from option F is false.

Option	Statement	True or False?
F	The standard deviation is less than the mean absolute deviation.	False $\times$
G	The standard deviation is greater than the mean absolute deviation.	True $✓$

Now let's determine whether the other options are true or false.

H and I

In options H and I, we are given the following statements.

Option	Statement	True or False?
H	The mean absolute deviation is greater than the range.	?
I	The range is less than the standard deviation.	?

Since both of these statements mention range, let's find the range of our data set! To do that, we need to identify the minimum and maximum lengths in the data set.

Length (ft)
$12$	$15$	$15$	$20$

We can see that that the minimum values is $12$ and the maximum value is $20.$ The range is the difference of these values. We found that the range is equal to $8.$ Earlier, we found that the mean absolute deviation is $2,$ which is less than the range. Therefore, the mean absolute deviation is not greater than the range.

Option	Statement	True or False?
H	The mean absolute deviation is greater than the range.	False $\times$
I	The range is less than the standard deviation.	?

Also, since the standard deviation is equal to only $2.9,$ the range is not less than the standard deviation.

Option	Statement	True or False?
H	The mean absolute deviation is greater than the range.	False $\times$
I	The range is less than the standard deviation.	False $\times$

Let's summarize our results!

Option	Statement	True or False?
F	The standard deviation is less than the mean absolute deviation.	False $\times$
G	The standard deviation is greater than the mean absolute deviation.	True $✓$
H	The mean absolute deviation is greater than the range.	False $\times$
I	The range is less than the standard deviation.	False $\times$